Northern Ireland group meeting: Multivariate frailty models based on time dependent logistic function.

The Northern Ireland group held a meeting on Wednesday, 20 March 2019 at 1pm in the Peter Froggatt Centre in Queen's University of Belfast. The speaker was Dr Milisa Bucknall of the University of Keele, UK.

Dr Bucknall introduced the Generalised Time Dependent (GTDL) survival model defined by the following hazard function:

where: α and λ (>0) are scalars and β is a vector of p parameters measuring the influence of the covariates 𝑥. This model does not have proportional hazards. She then argued that the roles of λ and β0 (the intercept) were interchangeable (aliased), and that β0 should be omitted, leading to the Canonical form of the survival distribution, the CTDL. 

Next, Dr Bucknall went on to the develop (a) a Gamma frailty version of the CTDL model and (b) the Weibull Gamma frailty model as a comparator, noting that the latter leads to the Lomax type II distribution (a Pareto Power distribution). However the CTDL-Gamma frailty distribution remains to be formally identified and, at the time of writing, is considered novel. Both models were then used to analyse data from the Northern Ireland Lung Cancer Study: a set of 855 incident cases followed for two years. For illustrative purposes survival was analysed in relation to age, sex and treatment. The fit was better using the Weibull-Gamma model (AIC= 4093.1) compared to the CTDL-Gamma model (AIC=4112.3). However, the interpretation of the covariates was broadly similar - treatment signicantly improved survival, age signicantly decreased survival and sex did not influence survival.

Bivariate extensions of the models were developed using the idea of correlated gamma frailty and Clayton's definition of the bivariate hazard function. The first models were based on shared frailty and Dr Bucknall obtained the bi-variate survival functions for the two models and, interestingly, began to explore the correlation structure between the times (T1, T2) and the gamma components (U1U2) in each model by means of a simulation study. There was good correspondence between corr(T1, T2) and corr(U1, U2) in the Weibull-Gamma model but much less in the CTDL-Gamma model, the latter supporting a wider range of possible relationships. Finally, Milisa investigated which shape parameters influenced corr(T1, T2) and quantified their inuence by fitting a second-order linear model in the shape parameters in each model using the estimates obtained in the simulation study.

The talk was well received and led to some insightful comments and questions. The complexity of the derivations (omitted from the presentation) was enquired about. It was considerable involving several pages of algebra. However, an immediate generalisation of the bivariate forms to multivariate forms was pointed out. The audience agreed that such an extension was tractable and thanked the speaker for a very interesting presentation and a useful addition to the survival modelling literature.

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